The Relation between Two Present Value Formulae


 Marylou Martin
 3 years ago
 Views:
Transcription
1 James Ciecka, Gary Skoog, and Gerald Martin The Relation between Two Present Value Formulae. Journal of Legal Economics 15(): pp The Relation between Two Present Value Formulae James E. Ciecka, Gary R. Skoog, and Gerald D. Martin * David Jones recently raised an interesting question on a forensic LISTSERV. He observed that a handheld calculator returned $4.684 for the present value of an ordinary annuity of $1 for 4.5 years when evaluated at a discount rate of.0. However, the present value of annual payments of $1 for 4.0 years is $ ; and the present value of a final payment of $.50 in 4.5 years is a total of $ The difference between $4.684 and $ is small, but the question is why the two present values differ. In general notation, the Jones question could be phrased as follows: Let n be an integer number of years, i denotes the discount rate (assumed to be greater than zero and less than or equal to one), and let 0< < 1denote a fraction of a year and the amount of the payment made in the fractional year. In Jones s question, n = 4, =.5, n + = 4.5, and i =.0. A handheld calculator computes (1) ( n+ ) (1 / i)[1 ] as distinct from () (1 / i)[1 ] +. n+ This note investigates the relation between formulae (1) and (). * Ciecka: Professor of Economics, Department of Economics, DePaul University, Chicago, IL. Skoog: Legal Econometrics, Inc., & Professor of Economics, Department of Economics, DePaul University, Chicago, IL. Martin: Professor Emeritus, Department of Finance, California State University, Fresno, CA. Ciecka, Skoog, and Martin: The Relation between Two Present Value Formulae 61
2 First, we observe, 1 (3) 1 + i>. From inequality (3), we have (4a) 1 + i> repeating (3) (4b) + i > 1 rearranging (4a) (4c) (1 ) n + n+ + i + i> 1 adding (1 i) n + + to both sides of (4b) n+ n n+ ( n+ ) (4d) [1 ] + i> [1 ] regrouping (4c) (4e) n ( n+ ) (1 / i)[1 ] + > (1 / i)[1 ] n+ multiplying (4d) by i 1 ( + ) Since the left side of (4e) is formula () and the right side is formula (1), we have established that formula () exceeds formula (1) for 0< < 1, 0< i 1, and positive integer values n as exemplified in the Jones question. The difference between formulae (1) and () can be approximated by expanding formulae (1) and () with the general binomial theorem. 1 This can be seen from the expansion of using the general binomial theorem which gives: 3 = 1 + i+ (1/ ) ( 1) i + (1/ 6) ( 1)( ) i +K., noting that the third term in the expansion is negative and the fourth is positive but smaller than the absolute value of the third term. Successive pairs of terms follow the same pattern. Therefore, = 1+ i+ [a negative amount], and 1 + i>. See Appendix 1 for this result. Journal of Legal Economics 6 Volume 15, Number, April 009, pp
3 (5) 1 (1 ) i n For example, in the Jones formulation, (5) evaluates to 1 (1 ) 1.5(1.5) (5a) i (.0).0031 n 4 = = (1 +.0) where the actual difference is.006 = The difference between formulae (1) and () is small (especially for net discount rates used in forensic work), but () does exceed (1). Formula (1) can be viewed as an extension of the formula for an annuity immediate but with a noninteger term rather than an integer term. One might think of it as equivalent to a level annuity paid at points in time n + ( ) 3( ) ( 1)( ), n +, n +,, n + n + K [i.e., at equal intervals n+ 1 n+ 1 n+ 1 n+ 1 of ( n + )/( n + 1) ]. We can then find the periodic payment that would be just sufficient to make the present value of payments equal to the value produced by formula (1). For example, in Jones s question, payments would be made at points in time (4 +.5) 3(4 +.5) 4(4 +.5) (5)(4 +.5),,,,, which simplifies to.9, 1.8,.7, 3.6, 4.5 years into the future. A level annuity of $ has a present value of $4.684 as results from formula (1); and, in that sense, is equivalent to formula (1). Figure 1 is the time diagram for this annuity. Figures a and b show time diagrams using formula () and an annuity equivalent to (), both of which equal $ Figure b shows slightly higher payments than Figure 1 at exactly the same points in time, resulting in a greater present value. This result is consistent with inequality (4e) which established that formula () produces a greater present value than formula (1). Ciecka, Skoog, and Martin: The Relation between Two Present Value Formulae 63
4 Figure 1. Present Value of Payments Equivalent to Formula 1 Present Value = $ Figure a. Present Value of Payments Using Formula Present Value = $ Figure b. Present Value of Payments Equivalent to Formula Present Value = $ We can say more if we view the two objects being compared as functions of a real variable, s, defined on the positive real numbers. Using the usual notation [s] to indicate the greatest integer in s, n = [s] and s = n+ = [] s +, we have, for all positive s, from (1), PVSS( s; i) = (1/ i)[1 s ] from () [ s] ( s [ s]) PVEXACT (;) s i (1/)[1 i ] + s. The notation PVSS is chosen to reflect the present value function embedded in commercial spreadsheets, in particular, in Microsoft Excel. In fact, the Excel function is PV (, i nper, pmt, fv, type ), with nper being the number of periods our s. Additionally, pmt, the payment per period, is 1, fv (which can convert the spreadsheet to a future value calculation) is set to 0, and type, which governs whether the payments occur at the beginning or the end of periods, is set to 0 to reflect our endofperiod assumption. Thus PVSS(;) s i PV (,,1,0,0) i s. In fact, the Microsoft help screen for PV discusses nper as if it were an integer it is not Journal of Legal Economics 64 Volume 15, Number, April 009, pp
5 even clear that Microsoft gave any thought to the problem being discussed here. Now because Excel has the [s] built in as its INT(s) function, the forensic economist wishing to avoid the small error being discussed here can avoid it by creating his own userdefined function within Excel using our formula for PVEXACT. The earlier discussion showed that PVEXACT (;) s i PVSS(;) s i, with equality on the integers and inequality off the integers. Both PVEXACT (;) s i and PVSS(;) s i are continuous, monotonically increasing functions on (0, ) which agree with ( s) (1 / i)[1 ] when s takes on the value of an integer n. PVSS is infinitely differentiable and, as taking two derivatives shows, everywhere strictly concave. PVEXACT is infinitely differentiable and concave only within any interval which contains no integers. It is not differentiable at any integer, nor is it concave in any interval containing an integer. These departures or failures result from its left hand derivative being less than its right hand derivative on the integers, so that its graph may be described as being the graph of PVSS s, but with concave arcs superimposed across the intervals between consecutive integers. As s increases, these arcs disappear in the limit, as illustrated in Figure 3 for i =.04 and s = 40 years. 3 In Appendix 3 we analytically compute these left and righthand derivatives, and relate them to the derivative of PVSS. 3 The smooth function is PVSS. The PVEXACT function consists of a series of cusps which coincide with PVSS on the integers. We exaggerated the bend in the cusps in order to better illustrate the PVEXACT function. Ciecka, Skoog, and Martin: The Relation between Two Present Value Formulae 65
6 Figure 3. PVSS and PVEXACT Functions with i =.04 The extension of the PVSS function from the integers to the real numbers follows an old tradition in mathematics: the principle of the permanence of form. The latter idea consists in extending the fundamental laws and operations which are applicable to positive integers to ever wider collections of numbers here rational and all irrational numbers. Mathematicians who developed this argument include Peacock, in his Arithmetic and Symbolic Algebra in 184, Hankel in his Complexe Zahli'imysfeme in 1867, and Cantor, extending results to the irrationals in Now, given that a function has its values prescribed on the positive integers, there are infinitely many functions which extend these values to the real numbers. Indeed, because the function m= k bm sin( π ms) for arbitrary choices of { b m } vanishes on integer m= 1 values of s, it may be added to any extension to produce another equally valid extension. Despite the appeal of the permanence of form argument, it clearly leads to the wrong extension of the integral present value function since it does not coincide with the economically meaningful PVEXACT function. 4 College Algebra, by James Harrington Boyd, Scott, Foresman and Company:1901. Journal of Legal Economics 66 Volume 15, Number, April 009, pp
7 As soon as we allow payments at arbitrary points in time between integers, it is natural to consider payments at all points in time between the integers, i.e. to consider continuous annuities. We then have t = s rt 1 rs rt e t= s e t= 0 r r t= 0 PVCTS(;) s i e dt = = r where e = 1/ so that r is the continuously compounded rate of interest corresponding to the annually compounded interest rate of i. Multiplication of the right hand side of PVCTS by i/ i results s i 1 i in PVCTS(;) s i = = PVSS(;) s i.this gives another r i r physical interpretation of PVSS, in addition to that offered earlier as involving equal payments of 1 at intervals ( n+ )/( n+ 1) apart. Here PVSS is shown to correspond to payments of a continuous annuity of ( ri / ) < 1, where the latter inequality follows from: r r e = 1+ r+ + L = 1+ i, so that i > r.! Of course, if the continuous annuity is 1, i > r implies that PVCTS>PVSS. Now, a continuous annuity speeds up the uniform payments as much as possible away from the end of the period, so we expect that, if the payment is the same, its present value will be larger. Our definition of PVCTS attempted to correct for this acceleration as much as possible, by taking its argument, i, and adjusting it downward to r. Ciecka, Skoog, and Martin: The Relation between Two Present Value Formulae 67
8 Appendix 1 The difference between formula () and (1) is (A1) (A) (A3) ( n+ ) (1 / i)[1 ] + (1 / i)[1 ] n+ ( n+ ) ( n+ ) (1/ i)[1 (1 i) i(1 i) 1 (1 i) ] = ( n+ ) ( n+ ) (1 / i)[(1 i) i(1 i) (1 i) ] = = (A4) (1/ i) [ + i 1] = (A5) (1/ i) [ 1] = (A6) ( 1) i + i i+ i + + i = n (1/ )(1 ) {[1...](1 ) 1} (A7) n ( + 1) ( + 1) 3 (1 / i) {1 + i i i + i + i } (A8) (A9) (A10) n ( + 1) (1 / )(1 ) [ ] i + i i + i = i + i + i (1 / i) [ ] = + i [ ] = (A11) 1 (1 ) i n repeating text formula (5) Steps (A)(A5) rearrange (A1). (A6) uses the general binomial theorem. (A7) is a rearrangement of (A6). Terms involving i 3 and higher order terms in i are dropped in (A8). Steps (A9)(A11) simplify (A8). We note that (A11) approaches zero as approaches either zero or one (i.e., the term of the annuity approaches integer Journal of Legal Economics 68 Volume 15, Number, April 009, pp
9 values of n or n + 1). Also, (A11) approaches zero as n approaches infinity and as i approaches zero. Ciecka, Skoog, and Martin: The Relation between Two Present Value Formulae 69
10 Appendix This note, along with Appendix 1, has provided the general notation to explain why there is a difference in the present value depending upon whether one is using a calculator, the Excel PVSS function, or making the calculations year by year in a spreadsheet. David Jones, in his query, posed a simple data set consisting of a 4.5 year time period, a % interest rate, and a constant $1.00 per year payout for 4 years followed by one payment of $0.50 at year 4.5. He thereby eliminated the need to consider any growth rate in the payout. However, one could easily envision the % interest rate as being a Net Discount Rate (NDR) which many economists use to calculate present value and in which the growth rate and interest rate are combined so that the growth rate can be assumed to be 0%. The Table in Appendix has been included so that the reader can visualize the difference as seen by Jones when he made the calculations and posed his question. These are the same variables as used in the general notation discussion in this note. The first section of the Table contains the variables entered into a handheld calculator. Using the variables listed in the preceding paragraph, the calculator yields a present value of $ This was verified by using three calculators by three different manufacturers to insure that all are using the same formula. That formula is provided in the first section of the Table. In the second section of the Table, the variables were entered into the PV (Present Value) function of an Excel spreadsheet (PVSS). The answer was found to be identical to that obtained using a calculator: $ The data entry sequence is different in Excel when compared to the data entry sequence in a calculator, but the underlying formulae are the same. To see the Excel formula, enter the Help section of Excel and then type PV Function in the search box. The third and final section of the Table is a spreadsheet wherein the same variables are used to create a year by year calculation of the present value, with the value for each year calculated individually and shown in the PV column. These are then summed to arrive at a present value of $ Each PV cell in the spreadsheet contains the formula for the present value of a lump sum, and that formula is shown above the spreadsheet. This note has established that the difference in the solutions is minor; therefore, either method could be used. As explained in the general notation discussion, this is not a constant difference and will change as the input variables change. However, we emphasize that Journal of Legal Economics 70 Volume 15, Number, April 009, pp
11 the difference will always be very small with the year by year spreadsheet solution always being slightly higher than the calculator solution or the Excel PV (PVSS) function solution. Handheld Calculator v. Excel PV Function (PVSS) v. Year by Year Calculation Using Handheld Calculator Formula =PMT*(1/i)(1/(i*(1+i)^n)) This is the formula used in handheld calculators. Years (n) 4.5 Interest (i) % Payment/Period(PMT) $1.00 PV $4.684 Using Excel PV Function (PVSS) Excel PV Function =PV(D19/D3,D0*D3,D1,D,D17) Col. D Row 17 End of Period 0 Row 18 Row 19 Int.0% Row 0 n (years) 4.5 Row 1 Pmt $1.00 Row FV $0.00 Row 3 m (discounting periods/yr) 1 PV $4.684 For explanation of formula see PV Function in Excel Help. Using PV Formula in Year by Year Calculation Formula for calculating the present value of a lump sum pv = fv / (1+i)^n This is the formula entered by user in each of the PV cells. n FV($) i PV $ $ $ $ $ Present value = $ Ciecka, Skoog, and Martin: The Relation between Two Present Value Formulae 71
12 Appendix 3 This appendix quantifies and proves the inequality among the derivatives (right and left) for PVEXACT (;) s i and PVSS(;) s i suggested in the text, and notes a lemma connecting the continuous and discrete interest rates which ties these together. First, the Lemma. For a given discrete interest rate i and its r continuous analogue r, related by e = we have, for i > 0, 1 r < r < 1. i Proof: Only the first inequality needs to be shown, since the second one has been established earlier. The first inequality is equivalent to i< r = r(1 + r+ r + L ). 1 r r r r But e = 1+ r+ + L = 1+ i i= r+ + L< r+ r L = r.!! (1 r) We now state the result which describes the behavior of the directional derivatives at the integers in the graph. Proposition. PVSS ( n; i) = ( r / i)( e rn ) PVEXACT ( n; i) = e r + rn PVEXACT ( n; i) = e (1 r) Before proving this, notice first that, from the lemma, PVEXACT ( n; i) < PVSS ( n; i) < PVEXACT + ( n; i). Further, all derivatives are positive, and all go to zero with increasing n. The proof is computation. The first derivative is easiest: s rs PVSS(;) s i (1/)[1 i (1 i) = + ] = (1/)[1 i e ]. rs Clearly PVSS (;) s i = ( r /)[ i e ] = ( r /) i 1 that on the integers, for arbitrary n, rn PVSS ( n; i) = ( r / i)[ e ] = ( r / i) 1 n. s everywhere, so Journal of Legal Economics 7 Volume 15, Number, April 009, pp
13 More difficult are the right hand and left hand derivatives (note the + and subscripts) at an integer n, given by definition as PVEXACT ( n + ; i) PVEXACT ( n; i) PVEXACT + ( n; i) lim 0 PVEXACT ( n; i) = lim 0 lim 0 PVEXACT ( n + ; i) PVEXACT ( n; i) PVEXACT ( n ; i) PVEXACT ( n; i) Starting with the easier PVEXACT + ( n; i), forming the numerator in its limit involves the next two terms: [ n+ ] ( n+ [ n+ ]) PVEXACT ( n + ; i) (1/ i)[1 ] + n+ = (1 / i)[1 ] + n+ [ n] ( n [ n]) PVEXACT ( n; i) (1/ i)[1 ] + = (1/ i)[1 ] n 1 1 r r PVEXACT + ( n; i) lim = = e > e n+ n 0 i = PVSS ( n; i) r Finally, the more difficult left limit: [ n ] ( n [ n ]) PVEXACT ( n ; i) (1/ i)[1 ] + n ( n 1) = (1 / i)[1 ] + n ( n 1) n r( n 1) r( n ) = (1 / i)[1 e ] + e (1 ) and ( n) ( n [ n]) rn PVEXACT ( n; i) (1/ i)[1 ] + = (1/ i)[1 e ] n where [ n ] = n 1. Ciecka, Skoog, and Martin: The Relation between Two Present Value Formulae 73
14 PVEXACT ( n; i) = lim 0 lim 0 r( n 1) r( n ) rn (1 / i)[1 e ] + e (1 ) (1 / i)[1 e ] + rn r( n 1) r( n ) (1 / i)[ e e ] e (1 ) r r( n ) rn (1 e ) e (1 ) = lim (1 / ie ) + 0 r rn r rn ( e 1) e e rn r = lim (1 / ) e + e e 0 i r 1 e = lim e + e e 0 r 1 e = e lim e lime rn = e (1 r) rn rn r rn rn r Of the seven equalities, the first substitutes the definition of left hand derivative and uses the fact that for [ n ] = n 1,for > 0, the second cancels 1 i terms, the third factors out rn e, the fourth regroups and redistributes, the fifth uses the previously noted r continuous interest/discrete interest equality e = 1 + i in the form ( r e 1) = 1, the sixth groups two terms which individually i would go to infinity, and the seventh uses the observation that 3 1 (1 ( ) ( ) 1 r )! r r r e L = 3! = r+ o( ) where o( ) m eans that the terms divided by the argument go to 0 faster than (L Hospital s Rule would work as well). Journal of Legal Economics 74 Volume 15, Number, April 009, pp
9.2 Summation Notation
9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a
More information8.7 Mathematical Induction
8.7. MATHEMATICAL INDUCTION 8135 8.7 Mathematical Induction Objective Prove a statement by mathematical induction Many mathematical facts are established by first observing a pattern, then making a conjecture
More informationSequences and Series
Contents 6 Sequences and Series 6. Sequences and Series 6. Infinite Series 3 6.3 The Binomial Series 6 6.4 Power Series 3 6.5 Maclaurin and Taylor Series 40 Learning outcomes In this Workbook you will
More informationANALYTICAL MATHEMATICS FOR APPLICATIONS 2016 LECTURE NOTES Series
ANALYTICAL MATHEMATICS FOR APPLICATIONS 206 LECTURE NOTES 8 ISSUED 24 APRIL 206 A series is a formal sum. Series a + a 2 + a 3 + + + where { } is a sequence of real numbers. Here formal means that we don
More informationMATHEMATICS OF FINANCE AND INVESTMENT
MATHEMATICS OF FINANCE AND INVESTMENT G. I. FALIN Department of Probability Theory Faculty of Mechanics & Mathematics Moscow State Lomonosov University Moscow 119992 g.falin@mail.ru 2 G.I.Falin. Mathematics
More informationBasic financial arithmetic
2 Basic financial arithmetic Simple interest Compound interest Nominal and effective rates Continuous discounting Conversions and comparisons Exercise Summary File: MFME2_02.xls 13 This chapter deals
More informationSECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS
(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic
More informationEXCEL PREREQUISITES SOLVING TIME VALUE OF MONEY PROBLEMS IN EXCEL
CHAPTER 3 Smart Excel Appendix Use the Smart Excel spreadsheets and animated tutorials at the Smart Finance section of http://www.cengage.co.uk/megginson. Appendix Contents Excel prerequisites Creating
More information2 A Differential Equations Primer
A Differential Equations Primer Homer: Keep your head down, follow through. [Bart putts and misses] Okay, that didn't work. This time, move your head and don't follow through. From: The Simpsons.1 Introduction
More informationLesson 1. Key Financial Concepts INTRODUCTION
Key Financial Concepts INTRODUCTION Welcome to Financial Management! One of the most important components of every business operation is financial decision making. Business decisions at all levels have
More information1 Interest rates, and riskfree investments
Interest rates, and riskfree investments Copyright c 2005 by Karl Sigman. Interest and compounded interest Suppose that you place x 0 ($) in an account that offers a fixed (never to change over time)
More informationSome Notes on Taylor Polynomials and Taylor Series
Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited
More information16.1. Sequences and Series. Introduction. Prerequisites. Learning Outcomes. Learning Style
Sequences and Series 16.1 Introduction In this block we develop the ground work for later blocks on infinite series and on power series. We begin with simple sequences of numbers and with finite series
More informationThe Basics of Interest Theory
Contents Preface 3 The Basics of Interest Theory 9 1 The Meaning of Interest................................... 10 2 Accumulation and Amount Functions............................ 14 3 Effective Interest
More informationSequences with MS
Sequences 20082014 with MS 1. [4 marks] Find the value of k if. 2a. [4 marks] The sum of the first 16 terms of an arithmetic sequence is 212 and the fifth term is 8. Find the first term and the common
More informationrate nper pmt pv Interest Number of Payment Present Future Rate Periods Amount Value Value 12.00% 1 0 $100.00 $112.00
In Excel language, if the initial cash flow is an inflow (positive), then the future value must be an outflow (negative). Therefore you must add a negative sign before the FV (and PV) function. The inputs
More informationINTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS
INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28
More informationVilnius University. Faculty of Mathematics and Informatics. Gintautas Bareikis
Vilnius University Faculty of Mathematics and Informatics Gintautas Bareikis CONTENT Chapter 1. SIMPLE AND COMPOUND INTEREST 1.1 Simple interest......................................................................
More informationCHAPTER 3 Numbers and Numeral Systems
CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,
More information6: Financial Calculations
: Financial Calculations The Time Value of Money Growth of Money I Growth of Money II The FV Function Amortisation of a Loan Annuity Calculation Comparing Investments Worked examples Other Financial Functions
More informationLectures 56: Taylor Series
Math 1d Instructor: Padraic Bartlett Lectures 5: Taylor Series Weeks 5 Caltech 213 1 Taylor Polynomials and Series As we saw in week 4, power series are remarkably nice objects to work with. In particular,
More informationIt is time to prove some theorems. There are various strategies for doing
CHAPTER 4 Direct Proof It is time to prove some theorems. There are various strategies for doing this; we now examine the most straightforward approach, a technique called direct proof. As we begin, it
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationA Curious Case of a Continued Fraction
A Curious Case of a Continued Fraction Abigail Faron Advisor: Dr. David Garth May 30, 204 Introduction The primary purpose of this paper is to combine the studies of continued fractions, the Fibonacci
More informationTIME VALUE OF MONEY. In following we will introduce one of the most important and powerful concepts you will learn in your study of finance;
In following we will introduce one of the most important and powerful concepts you will learn in your study of finance; the time value of money. It is generally acknowledged that money has a time value.
More informationSome Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.
Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,
More information3. Time value of money. We will review some tools for discounting cash flows.
1 3. Time value of money We will review some tools for discounting cash flows. Simple interest 2 With simple interest, the amount earned each period is always the same: i = rp o where i = interest earned
More informationTaylor and Maclaurin Series
Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions
More informationThe Time Value of Money
C H A P T E R6 The Time Value of Money When plumbers or carpenters tackle a job, they begin by opening their toolboxes, which hold a variety of specialized tools to help them perform their jobs. The financial
More information5. Time value of money
1 Simple interest 2 5. Time value of money With simple interest, the amount earned each period is always the same: i = rp o We will review some tools for discounting cash flows. where i = interest earned
More informationNotes on Factoring. MA 206 Kurt Bryan
The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor
More information3. Time value of money
1 Simple interest 2 3. Time value of money With simple interest, the amount earned each period is always the same: i = rp o We will review some tools for discounting cash flows. where i = interest earned
More informationCommon Curriculum Map. Discipline: Math Course: College Algebra
Common Curriculum Map Discipline: Math Course: College Algebra August/September: 6A.5 Perform additions, subtraction and multiplication of complex numbers and graph the results in the complex plane 8a.4a
More informationIndiana State Core Curriculum Standards updated 2009 Algebra I
Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and
More informationChapter 2 Limits Functions and Sequences sequence sequence Example
Chapter Limits In the net few chapters we shall investigate several concepts from calculus, all of which are based on the notion of a limit. In the normal sequence of mathematics courses that students
More informationA Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions
A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25
More informationPayment streams and variable interest rates
Chapter 4 Payment streams and variable interest rates In this chapter we consider two extensions of the theory Firstly, we look at payment streams A payment stream is a payment that occurs continuously,
More information2.1.1 Examples of Sets and their Elements
Chapter 2 Set Theory 2.1 Sets The most basic object in Mathematics is called a set. As rudimentary as it is, the exact, formal definition of a set is highly complex. For our purposes, we will simply define
More informationPURSUITS IN MATHEMATICS often produce elementary functions as solutions that need to be
Fast Approximation of the Tangent, Hyperbolic Tangent, Exponential and Logarithmic Functions 2007 Ron Doerfler http://www.myreckonings.com June 27, 2007 Abstract There are some of us who enjoy using our
More informationCalculations for Time Value of Money
KEATMX01_p001008.qxd 11/4/05 4:47 PM Page 1 Calculations for Time Value of Money In this appendix, a brief explanation of the computation of the time value of money is given for readers not familiar with
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationUNDERSTANDING HEALTHCARE FINANCIAL MANAGEMENT, 5ed. Time Value Analysis
This is a sample of the instructor resources for Understanding Healthcare Financial Management, Fifth Edition, by Louis Gapenski. This sample contains the chapter models, endofchapter problems, and endofchapter
More informationTime Value of Money. Background
Time Value of Money (Text reference: Chapter 4) Topics Background One period case  single cash flow Multiperiod case  single cash flow Multiperiod case  compounding periods Multiperiod case  multiple
More information2. How would (a) a decrease in the interest rate or (b) an increase in the holding period of a deposit affect its future value? Why?
CHAPTER 3 CONCEPT REVIEW QUESTIONS 1. Will a deposit made into an account paying compound interest (assuming compounding occurs once per year) yield a higher future value after one period than an equalsized
More informationCalculus for Middle School Teachers. Problems and Notes for MTHT 466
Calculus for Middle School Teachers Problems and Notes for MTHT 466 Bonnie Saunders Fall 2010 1 I Infinity Week 1 How big is Infinity? Problem of the Week: The Chess Board Problem There once was a humble
More informationThis is Time Value of Money: Multiple Flows, chapter 7 from the book Finance for Managers (index.html) (v. 0.1).
This is Time Value of Money: Multiple Flows, chapter 7 from the book Finance for Managers (index.html) (v. 0.1). This book is licensed under a Creative Commons byncsa 3.0 (http://creativecommons.org/licenses/byncsa/
More informationALGEBRA 2 CRA 2 REVIEW  Chapters 16 Answer Section
ALGEBRA 2 CRA 2 REVIEW  Chapters 16 Answer Section MULTIPLE CHOICE 1. ANS: C 2. ANS: A 3. ANS: A OBJ: 53.1 Using Vertex Form SHORT ANSWER 4. ANS: (x + 6)(x 2 6x + 36) OBJ: 64.2 Solving Equations by
More informationArithmetic and Geometric Sequences
Arithmetic and Geometric Sequences Felix Lazebnik This collection of problems is for those who wish to learn about arithmetic and geometric sequences, or to those who wish to improve their understanding
More informationChapter 21: The Discounted Utility Model
Chapter 21: The Discounted Utility Model 21.1: Introduction This is an important chapter in that it introduces, and explores the implications of, an empirically relevant utility function representing intertemporal
More informationALGEBRA I A PLUS COURSE OUTLINE
ALGEBRA I A PLUS COURSE OUTLINE OVERVIEW: 1. Operations with Real Numbers 2. Equation Solving 3. Word Problems 4. Inequalities 5. Graphs of Functions 6. Linear Functions 7. Scatterplots and Lines of Best
More informationThe NotFormula Book for C1
Not The NotFormula Book for C1 Everything you need to know for Core 1 that won t be in the formula book Examination Board: AQA Brief This document is intended as an aid for revision. Although it includes
More informationHow to calculate present values
How to calculate present values Back to the future Chapter 3 Discounted Cash Flow Analysis (Time Value of Money) Discounted Cash Flow (DCF) analysis is the foundation of valuation in corporate finance
More informationReal Numbers and Monotone Sequences
Real Numbers and Monotone Sequences. Introduction. Real numbers. Mathematical analysis depends on the properties of the set R of real numbers, so we should begin by saying something about it. There are
More informationPV Tutorial Using Excel
EYK 153 PV Tutorial Using Excel TABLE OF CONTENTS Introduction Exercise 1: Exercise 2: Exercise 3: Exercise 4: Exercise 5: Exercise 6: Exercise 7: Exercise 8: Exercise 9: Exercise 10: Exercise 11: Exercise
More informationMAT2400 Analysis I. A brief introduction to proofs, sets, and functions
MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take
More informationLimit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x)
SEC. 4.1 TAYLOR SERIES AND CALCULATION OF FUNCTIONS 187 Taylor Series 4.1 Taylor Series and Calculation of Functions Limit processes are the basis of calculus. For example, the derivative f f (x + h) f
More informationMath Workshop October 2010 Fractions and Repeating Decimals
Math Workshop October 2010 Fractions and Repeating Decimals This evening we will investigate the patterns that arise when converting fractions to decimals. As an example of what we will be looking at,
More informationWHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?
WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly
More information31 is a prime number is a mathematical statement (which happens to be true).
Chapter 1 Mathematical Logic In its most basic form, Mathematics is the practice of assigning truth to welldefined statements. In this course, we will develop the skills to use known true statements to
More informationHow To Use Excel To Compute Compound Interest
Excel has several built in functions for working with compound interest and annuities. To use these functions, we ll start with a standard Excel worksheet. This worksheet contains the variables used throughout
More informationMATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear:
MATH 89 PROBLEM SET : INDUCTION The induction Principle The following property of the natural numbers is intuitively clear: Axiom Every nonempty subset of the set of nonnegative integers Z 0 = {0,,, 3,
More informationThe program also provides supplemental modules on topics in geometry and probability and statistics.
Algebra 1 Course Overview Students develop algebraic fluency by learning the skills needed to solve equations and perform important manipulations with numbers, variables, equations, and inequalities. Students
More informationIng. Tomáš Rábek, PhD Department of finance
Ing. Tomáš Rábek, PhD Department of finance For financial managers to have a clear understanding of the time value of money and its impact on stock prices. These concepts are discussed in this lesson,
More informationBond Price Arithmetic
1 Bond Price Arithmetic The purpose of this chapter is: To review the basics of the time value of money. This involves reviewing discounting guaranteed future cash flows at annual, semiannual and continuously
More informationCHAPTER 4. The Time Value of Money. Chapter Synopsis
CHAPTER 4 The Time Value of Money Chapter Synopsis Many financial problems require the valuation of cash flows occurring at different times. However, money received in the future is worth less than money
More informationLecture Notes on the Mathematics of Finance
Lecture Notes on the Mathematics of Finance Jerry Alan Veeh February 20, 2006 Copyright 2006 Jerry Alan Veeh. All rights reserved. 0. Introduction The objective of these notes is to present the basic aspects
More informationDick Schwanke Finite Math 111 Harford Community College Fall 2013
Annuities and Amortization Finite Mathematics 111 Dick Schwanke Session #3 1 In the Previous Two Sessions Calculating Simple Interest Finding the Amount Owed Computing Discounted Loans Quick Review of
More informationVISUAL ALGEBRA FOR COLLEGE STUDENTS. Laurie J. Burton Western Oregon University
VISUAL ALGEBRA FOR COLLEGE STUDENTS Laurie J. Burton Western Oregon University VISUAL ALGEBRA FOR COLLEGE STUDENTS TABLE OF CONTENTS Welcome and Introduction 1 Chapter 1: INTEGERS AND INTEGER OPERATIONS
More informationBookTOC.txt. 1. Functions, Graphs, and Models. Algebra Toolbox. Sets. The Real Numbers. Inequalities and Intervals on the Real Number Line
College Algebra in Context with Applications for the Managerial, Life, and Social Sciences, 3rd Edition Ronald J. Harshbarger, University of South Carolina  Beaufort Lisa S. Yocco, Georgia Southern University
More information5 =5. Since 5 > 0 Since 4 7 < 0 Since 0 0
a p p e n d i x e ABSOLUTE VALUE ABSOLUTE VALUE E.1 definition. The absolute value or magnitude of a real number a is denoted by a and is defined by { a if a 0 a = a if a
More informationCORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA
We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical
More informationImportant Financial Concepts
Part 2 Important Financial Concepts Chapter 4 Time Value of Money Chapter 5 Risk and Return Chapter 6 Interest Rates and Bond Valuation Chapter 7 Stock Valuation 130 LG1 LG2 LG3 LG4 LG5 LG6 Chapter 4 Time
More informationRafal Borkowski, Hipoteczna 18/22 m. 8, 91337 Lodz, POLAND, Email: rborkowski@go2.pl
Rafal Borkowski, Hipoteczna 18/22 m. 8, 91337 Lodz, POLAND, Email: rborkowski@go2.pl Krzysztof M. Ostaszewski, Actuarial Program Director, Illinois State University, Normal, IL 617904520, U.S.A., email:
More informationLesson 4 Annuities: The Mathematics of Regular Payments
Lesson 4 Annuities: The Mathematics of Regular Payments Introduction An annuity is a sequence of equal, periodic payments where each payment receives compound interest. One example of an annuity is a Christmas
More informationSenior Secondary Australian Curriculum
Senior Secondary Australian Curriculum Mathematical Methods Glossary Unit 1 Functions and graphs Asymptote A line is an asymptote to a curve if the distance between the line and the curve approaches zero
More informationMathematics of Cryptography
CHAPTER 2 Mathematics of Cryptography Part I: Modular Arithmetic, Congruence, and Matrices Objectives This chapter is intended to prepare the reader for the next few chapters in cryptography. The chapter
More information1. Periodic Fourier series. The Fourier expansion of a 2πperiodic function f is:
CONVERGENCE OF FOURIER SERIES 1. Periodic Fourier series. The Fourier expansion of a 2πperiodic function f is: with coefficients given by: a n = 1 π f(x) a 0 2 + a n cos(nx) + b n sin(nx), n 1 f(x) cos(nx)dx
More informationTaylor Polynomials and Taylor Series Math 126
Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will
More informationModule 5: Interest concepts of future and present value
file:///f /Courses/201011/CGA/FA2/06course/m05intro.htm Module 5: Interest concepts of future and present value Overview In this module, you learn about the fundamental concepts of interest and present
More informationMultistate transition models with actuarial applications c
Multistate transition models with actuarial applications c by James W. Daniel c Copyright 2004 by James W. Daniel Reprinted by the Casualty Actuarial Society and the Society of Actuaries by permission
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More information2. THE xy PLANE 7 C7
2. THE xy PLANE 2.1. The Real Line When we plot quantities on a graph we can plot not only integer values like 1, 2 and 3 but also fractions, like 3½ or 4¾. In fact we can, in principle, plot any real
More informationThe time value of money: Part II
The time value of money: Part II A reading prepared by Pamela Peterson Drake O U T L I E 1. Introduction 2. Annuities 3. Determining the unknown interest rate 4. Determining the number of compounding periods
More informationCompounding Quarterly, Monthly, and Daily
126 Compounding Quarterly, Monthly, and Daily So far, you have been compounding interest annually, which means the interest is added once per year. However, you will want to add the interest quarterly,
More informationSection 1. Statements and Truth Tables. Definition 1.1: A mathematical statement is a declarative sentence that is true or false, but not both.
M3210 Supplemental Notes: Basic Logic Concepts In this course we will examine statements about mathematical concepts and relationships between these concepts (definitions, theorems). We will also consider
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More informationChapter 2. CASH FLOW Objectives: To calculate the values of cash flows using the standard methods.. To evaluate alternatives and make reasonable
Chapter 2 CASH FLOW Objectives: To calculate the values of cash flows using the standard methods To evaluate alternatives and make reasonable suggestions To simulate mathematical and real content situations
More information1. Annuity a sequence of payments, each made at equally spaced time intervals.
Ordinary Annuities (Young: 6.2) In this Lecture: 1. More Terminology 2. Future Value of an Ordinary Annuity 3. The Ordinary Annuity Formula (Optional) 4. Present Value of an Ordinary Annuity More Terminology
More informationMATH REVIEW KIT. Reproduced with permission of the Certified General Accountant Association of Canada.
MATH REVIEW KIT Reproduced with permission of the Certified General Accountant Association of Canada. Copyright 00 by the Certified General Accountant Association of Canada and the UBC Real Estate Division.
More informationCHAPTER 1. Compound Interest
CHAPTER 1 Compound Interest 1. Compound Interest The simplest example of interest is a loan agreement two children might make: I will lend you a dollar, but every day you keep it, you owe me one more penny.
More informationMATH 90 CHAPTER 1 Name:.
MATH 90 CHAPTER 1 Name:. 1.1 Introduction to Algebra Need To Know What are Algebraic Expressions? Translating Expressions Equations What is Algebra? They say the only thing that stays the same is change.
More informationTYPES OF NUMBERS. Example 2. Example 1. Problems. Answers
TYPES OF NUMBERS When two or more integers are multiplied together, each number is a factor of the product. Nonnegative integers that have exactly two factors, namely, one and itself, are called prime
More informationChapter 8. 48 Financial Planning Handbook PDP
Chapter 8 48 Financial Planning Handbook PDP The Financial Planner's Toolkit As a financial planner, you will be doing a lot of mathematical calculations for your clients. Doing these calculations for
More informationMathematical Induction with MS
Mathematical Induction 200814 with MS 1a. [4 marks] Using the definition of a derivative as, show that the derivative of. 1b. [9 marks] Prove by induction that the derivative of is. 2a. [3 marks] Consider
More informationIntroduction to the HewlettPackard (HP) 10BII Calculator and Review of Mortgage Finance Calculations
Introduction to the HewlettPackard (HP) 10BII Calculator and Review of Mortgage Finance Calculations Real Estate Division Sauder School of Business University of British Columbia Introduction to the HewlettPackard
More information8.3. GEOMETRIC SEQUENCES AND SERIES
8.3. GEOMETRIC SEQUENCES AND SERIES What You Should Learn Recognize, write, and find the nth terms of geometric sequences. Find the sum of a finite geometric sequence. Find the sum of an infinite geometric
More informationMAT116 Project 2 Chapters 8 & 9
MAT116 Project 2 Chapters 8 & 9 1 81: The Project In Project 1 we made a loan workout decision based only on data from three banks that had merged into one. We did not consider issues like: What was the
More informationTools of Algebra. Solving Equations. Solving Inequalities. Dimensional Analysis and Probability. Scope and Sequence. Algebra I
Scope and Sequence Algebra I Tools of Algebra CLE 3102.1.1, CFU 3102.1.10, CFU 3102.1.9, CFU 3102.2.1, CFU 3102.2.2, CFU 3102.2.7, CFU 3102.2.8, SPI 3102.1.3, SPI 3102.2.3, SPI 3102.4.1, 12 Using Variables,
More informationIf I offered to give you $100, you would probably
File C596 June 2013 www.extension.iastate.edu/agdm Understanding the Time Value of Money If I offered to give you $100, you would probably say yes. Then, if I asked you if you wanted the $100 today or
More information